How many noisy quantum copies do you need to distill purity?

Given n copies of a noisy quantum state, how many do you need before you can reliably reconstruct its purest component? Previous answers only worked asymptotically—as n grows to infinity. This paper gives a concrete, finite formula: the required copies scale with the desired output quality and the gap between the two largest eigenvalues, but not at all with the dimension of the state. The proof rests on combinatorics of random Young diagrams, a classical tool finding unexpected use in quantum resource theory.